Arbitrary Ring Research Articles

N-weak injective and n-weak flat modules under m-semidualizing bimodules

Let [Formula: see text] and [Formula: see text] be rings, the [Formula: see text] two non-negative integers and [Formula: see text] an (faithfully) [Formula: see text]-semidualizing bimodule. In this paper, we introduce, via special super finitely presented module [Formula: see text] with respect [Formula: see text], the concepts of [Formula: see text]-[Formula: see text]-weak injective and [Formula: see text]-[Formula: see text]-weak flat modules, and then we obtain some characterizations of two classes of these modules, namely [Formula: see text] and [Formula: see text] which are larger than that of [Formula: see text]-weak injective and [Formula: see text]-weak flat modules, respectively. Also, we investigate Foxby equivalence relative to these modules. Then over any arbitrary ring, we show some nice properties from the classes [Formula: see text] and [Formula: see text].

When quotient of a polynomial or power series ring by a monomial ideal is half-factorial

Suppose that K is a field, S = K [ x 1 , … , x n ] or S = K [ [ x 1 , … , x n ] ] . We present a characterization of those monomial ideals I of S, for which S/I is a half-factorial ring. This characterization is related to the structure of the base field K and also depends on the combinatorics of a graph constructed from the monomial ideal I. We also show that if R [ x ] or R [ [ x ] ] is half-factorial for an arbitrary commutative ring R, then R is an integral domain.

Rigid integral representations of quivers over arbitrary commutative rings

Rigid integral representations of quivers over arbitrary commutative rings

On the equation $a^{2}x=a$ in unital rings

In an arbitrary ring, the equation in the title defines the left strongly regular elements. Elements that are both left and right strongly regular are simply called strongly regular. If all elements of a ring are left strongly regular, then they are, in fact, strongly regular and this is the definition of strongly regular rings. We provide a characterization of when a left strongly regular element is indeed strongly regular, based on an intrinsic condition. While we show that it is not possible to give $2\times 2$ non-examples over $\mathbb{Z}$ or in certain matrix rings over $\mathbb{Z}_{n}$ for $n\in \{8,9,16\}$, we present two examples by George Bergman: a left strongly regular element that is not regular and a regular, left strongly regular element that is not strongly regular. Further, we prove results for left strongly regular (square) matrices over various types of rings and propose a conjecture, strongly supported by computational evidence: over commutative rings, left strongly regular matrices are strongly regular.

Effective generic freeness and applications to local cohomology

AbstractLet be a Noetherian domain and be a finitely generated ‐algebra. We study several features regarding the generic freeness over of an ‐module. For an ideal , we show that the local cohomology modules are generically free over under certain settings where is a smooth ‐algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over of a finitely generated ‐module.

Weil cohomology theories and their motivic Hopf algebroids

In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the theory of motives, we give three equivalent viewpoints on Weil cohomology theories: as a cohomology theory on smooth varieties, as a motivic spectrum and as a realization functor. We also associate to every Weil cohomology theory a motivic Hopf algebroid generalizing the construction we gave in Ayoub (2014) for the Betti cohomology. Exploiting results and constructions from Ayoub (2020), we are able to prove that the motivic Hopf algebroids of all the classical Weil cohomology theories are connective. In particular, they give rise to motivic Galois groupoids which are spectral affine groupoid schemes.

Rings and C*-algebras generated by commutators

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number N such that every element is a sum of N products of pairs of commutators. We show that one can take N≤2 for matrix rings, and that one may choose N≤3 for rings that contain a direct sum of matrix rings – this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that N≤6 can be arranged.For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. Using that a C*-algebra cannot be a radical extension over a proper ideal, we deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

On the subalgebra of invariant elements: Finiteness and immersions

Given the n-dimensional affine space over an arbitrary commutative ring k, G a group scheme flat and finitely presented over k and X⊂Akn a G-invariant affine and closed subscheme, we prove that the GIT quotient ▪ is of finite type over k, even if k is not noetherian, provided G is linearly reductive. This is well-known if ▪ is faithfully flat, which does not hold in general. We also explore the infinite-dimensional case. Concretely, we consider a G−k finitely presented projective module M and an arbitrary k-module N. We prove, under certain conditions on k and G, that the degrees of the generators of (S•(M∨⊗N))G and the degrees of the generators of the ideal of relations are bounded. We encode this property into the notion of partially generated graded (pgg) algebra and we give their main properties. In particular, we prove the existence of canonical equivariant immersions of spectra of pgg algebras in certain projective spaces.

Descent of splendid Rickard equivalences in alternating groups

We show that any block of an alternating group with an abelian defect group over an arbitrary complete discrete valuation ring is splendidly Rickard equivalent to its Brauer correspondent. This provides new evidence for a refined version of Broué's abelian defect group conjecture proposed by Kessar and Linckelmann.

Dynamic and static dipole polarizability of an Aharonov–Bohm ring

The properties of nanoscale rings are sensitive to Aharonov–Bohm fluxes threading the ring. In particular, neighboring angular momentum eigenstates become degenerate in cases of half-integer flux. Such degeneracies have a profound effect on the dipole polarizability leading to a divergent response. We analyze circular finite-width nanorings and derive a simple and compact expression for the polarizability valid for arbitrary ring geometry, frequency and magnetic flux. The dipole response is significantly reduced by finite-width effects, yet divergencies at half-integer flux are predicted irrespective of width. Finally, a non-divergent (but large) response is restored through a non-perturbative treatment of both magnetic and electric fields.

Flat model structures and Gorenstein objects in functor categories

We construct a flat model structure on the category ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ of additive functors from a small preadditive category $\mathcal {Q}$ satisfying certain conditions to the module category ${_{R}\mathsf {Mod}}$ over an associative ring $R$ , whose homotopy category is the $\mathcal {Q}$ -shaped derived category introduced by Holm and Jørgensen. Moreover, we prove that for an arbitrary associative ring $R$ , an object in ${_{\mathcal {Q},\,R}\mathsf {Mod}}$ is Gorenstein projective (resp., Gorenstein injective, Gorenstein flat, projective coresolving Gorenstein flat) if and only if so is its value on each object of $\mathcal {Q}$ , and hence improve a result by Dell'Ambrogio, Stevenson and Šťovíček.

$\mathscr{T}$-Commuting Generalized Derivations on Ideals and Semi-Prime Ideal

This paper's primary goal is to look at a quotient ring $\mathscr{A}/\mathscr{T}$ structure, where $\mathscr{A}$ is an arbitrary ring and $\mathscr{T}$ is a semi-prime ideal of $\mathscr{A}$. More precisely, we examine the differential identities in a semi-prime ideal of an arbitrary ring involving $\mathscr{T}$-commuting generalized derivation. Furthermore, examples are given to prove that the restrictions imposed on the hypothesisof the various theorems were not superfluous.

Reifying dynamical algebra: Maximal ideals in countable rings, constructively

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as “implies 0 = 1”) we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull’s lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.11 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders.

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups Gπ(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1.We establish a variant of double centralizer theorem for elementary unipotents xα(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup Xα, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in Gπ(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.

A criterion for the properness of star-transposition as an involution on n × n matrices over any associative star-ring

On the ring M n ( C ) of all n × n complex matrices A, the conjugate transpose involution A ↦ A * has the “properness” property A * A = 0 ⇒ A = 0 , as required for the existence of the partial order known as the “ * -order” on M n ( C ) . More generally, related questions are asked and answered about the ring M n ( R ) of all n × n matrices over an arbitrary associative ring R with any given involution R → R (as a generalization of the conjugacy map C → C ), yielding a corresponding “ * -transposition” involution on M n ( R ) . A criterion is found for this involution of M n ( R ) to be proper, so that M n ( R ) has a corresponding * -order. The special case R = Z k is also considered.

Matrices having nonzero outer inverses

It is well known that every nonzero von Neumann regular $m\times n$-matrix $A$ over an arbitrary ring $R$ has a nonzero outer inverse $n\times m$-matrix $B$ in the sense that $B=BAB$. Generalizing previous work on von Neumann regular matrices, the matrices having nonzero outer inverses over semiperfect rings are characterized as the matrices having some entry outside the Jacobson radical of $R$. Such matrices over finite semiperfect rings and finite commutative rings are counted, and several applications are given.

Units, zero-divisors and idempotents in rings graded by torsion-free groups

Abstract The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures.

The isomorphism problem for basic modules and the divisibility profile of the algebra of polynomials

While mutual congeniality of bases is known to guarantee that basic modules from so-related bases are isomorphic, the question of what can be said about isomorphism of basic modules in general has remained open. We show that, for some algebras, basic modules may be non-isomorphic. We also show that it is possible, for some algebras, for all basic modules to be isomorphic, regardless of congeniality. In the process, and as a byproduct, we introduce the notion of domains of divisibility of modules over arbitrary rings. The mechanism employed here to differentiate non-isomorphic basic modules is by showing that they have different domains of divisibility. Domains of divisibility measure how divisible a module can be; for those cases when divisibility is equivalent to injectivity, domains of divisibility provide a way to gauge injectivity of modules as an alternative to the domains of injectivity and other mechanisms in the literature. We focus on the algebra of polynomials with one variable, and observe that F[x] has a full divisibility profile for any field F. When F is algebraically closed, we see that the direct product and the direct sum of all Pascal basic modules have the smallest divisibility domain. We also analyze the diversity of the family of basic modules as we explore how many of those divisibility domains correspond to basic modules. We show that, for an arbitrary infinite field F, F[x] has infinitely many pairwise non-isomorphic basic modules, the divisibility profile of F[x] is complete, and for an algebraically closed field F, the collection of basic modules has any subset S of F with a non-empty at most countable complement, the set {x+α|α∉S}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{ x + \\alpha | \\alpha \ otin S \\}$$\\end{document} as a domain of divisibility for some basic F[x]-module.

Dimension of uncountably generated submodules

In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated Noetherian dimension of an $R$-module, where $R$ is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension and Noetherian dimension.
 They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules.
 It is proved that a quotient finite dimensional module $M$ has uncountably generated Krull dimension if and only if it has Krull dimension, but
 the values of these dimensions might differ.
 Similarly, a quotient finite dimensional module $M$ has uncountably generated Noetherian dimension if and only if it has Noetherian dimension.
 We also show that the Noetherian dimension of a quotient finite dimensional module $M$ with uncountably generated Noetherian dimension $\beta$ is less than or equal to $\omega _{1}+\beta $, where $\omega_{1}$ is the first uncountable ordinal number.

Equivariant resolutions over Veronese rings

AbstractWorking in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules .